On vertex-transitive, non-Cayley graphs of order pqr Original Research Article

In 1983, D. Marušič initiated the determination of the set NC of non-Cayley numbers. A number n belongs to NC if there exists a vertex-transitive, non-Cayley graph of order n. The status of all non-square-free numbers and the case when n is the product of two primes was settled recently by B.D. McKay and C.E. Praeger. Here we deal with the smallest unsolved case, when n is the product of three distinct odd primes. We list a set of numbers n of this form which belong to NC. We also show that if there exists a vertex-primitive or quasiprimitive non-Cayley graph of order n = pqr then the number n occurs on our list. Moreover, we conjecture that the list we compiled contains all non-Cayley numbers of the form n = pqr.